Kernel Density Approximation (KDA)
The kernel density approximation technique allows us to represent an arbitrary probability density function with a mixture of Gaussians.
In this framework, number, weight, mean, and covariance of each Gaussian component are automatically determined by mean-shift mode finding algorithm and curvature fitting.
The representation is as simple and efficient as parametric methods (e.g., Gaussian, or mixture of Gaussians) and as flexible as non-parametric techniques (e.g., kernel density estimation).
It is also very convenient for updating the density function sequentially, so very useful for real-time applications.
We applied this technique to adaptive background modeling, object tracking (including on-line target appearance modeling) and clustering (image segmentation, clustering for DNA microarray data).
Figure 1: Comparison between KDE and KDA. (top) density function constructed by KDE with 200 Gaussian kernels (bottom) its approximation by KDA with 5 Gaussian components (The number of component is determined automatically.)
We applied this technique to adaptive background modeling, object tracking (including on-line target appearance modeling) and clustering (image segmentation, clustering for DNA microarray data).
Figure 1: Comparison between KDE and KDA. (top) density function constructed by KDE with 200 Gaussian kernels (bottom) its approximation by KDA with 5 Gaussian components (The number of component is determined automatically.)
Kernel-Based Bayesian Filtering
Particle filters are powerful method to estimate the state in non-Gaussian/non-linear systems.
However, they frequently suffer from degeneracy or loss of diversity problem, especially in high dimensional cases, so they require a large number of samples or occasional re-sampling step to alleviate these problems.
To overcome the limitation, we proposed a new kernel-based Bayesian filtering technique based on analytical density functions -- Gaussian mixtures.
The main advantage of maintaining an analytical representation is efficient sampling, especially, in high dimensional problems so that the number of samples required for robust estimation is reduced.
The most challenging step to construct Bayesian filtering based on Gaussian mixtures is to avoid the exponential increase of the number of components in the posterior. We integrate KDA to simplify the density function after the multiplication of prediction and measurement densities, and maintain a compact density representation.
This framework is applied to build general object tracking algorithm, our method shows better performance than the SIR particle filtering. Since this technique requires a relatively small number of samples for the observation, it can be applied to many applications in which a large number of samples is not available or the measurement process is very expensive.
Figure 2: Kernel-based Bayesian filtering (1) prior (2) density after prediction by unscented transformation (3) measurement density by density interpolation (4) posterior by kernel density approximation
The most challenging step to construct Bayesian filtering based on Gaussian mixtures is to avoid the exponential increase of the number of components in the posterior. We integrate KDA to simplify the density function after the multiplication of prediction and measurement densities, and maintain a compact density representation.
This framework is applied to build general object tracking algorithm, our method shows better performance than the SIR particle filtering. Since this technique requires a relatively small number of samples for the observation, it can be applied to many applications in which a large number of samples is not available or the measurement process is very expensive.
Figure 2: Kernel-based Bayesian filtering (1) prior (2) density after prediction by unscented transformation (3) measurement density by density interpolation (4) posterior by kernel density approximation
Probabilistic Sensor Fusion
Even though sensor fusion techniques based on particle filters have been applied to object tracking, their implementations have been limited to combining likelihoods from multiple sensors.
Therefore, the number of observations is increased as many times as the number of sensors, because every sensor is involved in the observation for each sample.
Also, the multi-modality induced by the different characteristics of multiple sensors may be lost due to the limitations of current sensor fusion techniques based on particle filters.
We describe a methodology to model interactions between multiple sensors and to estimate the current state by using a mixture of Bayesian filters - one filter for each sensor, where each filter makes a different level of contribution to the posterior distribution. In this framework, an adaptive particle arrangement system is constructed in which a different number of samples is assigned to each sensor using prior distribution and partial observations. Also, multi-modality in the posterior is naturally preserved by the mixture model, which is realized by the mixture kernel-based Bayesian filtering.
Figure 3: Probabilistic sensor fusion framework with Bayesian filtering
We describe a methodology to model interactions between multiple sensors and to estimate the current state by using a mixture of Bayesian filters - one filter for each sensor, where each filter makes a different level of contribution to the posterior distribution. In this framework, an adaptive particle arrangement system is constructed in which a different number of samples is assigned to each sensor using prior distribution and partial observations. Also, multi-modality in the posterior is naturally preserved by the mixture model, which is realized by the mixture kernel-based Bayesian filtering.
Figure 3: Probabilistic sensor fusion framework with Bayesian filtering
Feature Extraction and Selection
Selecting or extracting a low-dimensional discriminative feature set is a critical step to improve the performance of tracking algorithms.
For object tracking, multiple heterogeneous features are assembled and likelihood images are constructed for various subspaces of the combined feature space.
Then, the most discriminative feature, which is extracted by PCA based on those likelihood images and adapted by incremental subspace update, is employed for object tracking.
